Ai Symbolic Neuro

Symbolic AI & Neuro-Symbolic Integration

Formal knowledge representation, symbolic reasoning, and their integration with neural methods for knowledge structuring and consolidation pipelines.

---

Routing

• USE FOR: Symbolic AI, neuro-symbolic integration, knowledge representation and reasoning (KRR), ontology engineering, description logics, and hybrid symbolic-neural architectures. Use for reasoning about formal knowledge structures, rule-based systems, logic programming, semantic web technologies, and their integration with neural approaches in knowledge graphs and structured reasoning pipelines.
• DO NOT USE FOR: formalising the ontology in Lean (use @lean-knowledge-formalization); commonsense reasoning (use @ai-commonsense-reasoning); agentic AI (use @ai-agentic-evolving).
• TRIGGERS: symbolic AI, neuro-symbolic, knowledge representation, KRR, ontology engineering, description logic, hybrid AI.

Workflow

1. Identify the integration question: symbolic-only, neural-only, or hybrid; pick the integration pattern (loose / tight / unified) from the body.
2. Pick the KRR formalism (description logic, RDF/OWL, conceptual graph, neural-symbolic embedding) that matches the application.
3. Apply the formalism; document where neural and symbolic components communicate + their semantic guarantees.
4. Hand off: to @lean-knowledge-formalization for Lean encoding, to @math-graph-knowledge for graph-theoretic foundations, to @lean-zettelkasten.

Recovery & STOP

• STOP if the question is purely commonsense-reasoning — delegate to @ai-commonsense-reasoning.
• STOP if the question is about causal/deontic logic — delegate to @ai-causal-deontic.
• STOP if the neural component requires empirical evaluation beyond methodological reasoning — escalate to @research-council.

Handoffs

• Predecessors: agent:gateway, skill:lean-research.
• Successors: skill:lean-knowledge-formalization, skill:ai-commonsense-reasoning, skill:math-graph-knowledge.

---

Part 1 — Knowledge Representation Foundations

1.1 Representation Languages

| Language | Expressiveness | Decidability | Common relevance | |---|---|---|---| | Propositional logic | Low | P (SAT: NP-complete) | Gate predicates and finite checks | | First-order logic (FOL) | High | Semi-decidable | Theorem formalization | | Description logic (ALC) | Medium | ExpTime | Ontology TBoxes | | OWL-DL | Medium-high | 2NExpTime | Semantic web ontologies | | Datalog | Limited FOL | P (data complexity) | Recursive queries on knowledge graphs | | Answer Set Programming | Nonmonotonic | ΣP2-complete | Default reasoning | | Modal logic | FOL + modalities | Varies | Epistemic/deontic reasoning |

1.2 Ontology Engineering

Foundational ontologies:

• DOLCE (Descriptive Ontology for Linguistic and Cognitive Engineering)
• BFO (Basic Formal Ontology) — ISO 21838
• SUMO (Suggested Upper Merged Ontology)

Design patterns:

• Parthood and mereology
• Temporal entities (endurants vs perdurants)
• Roles and role-playing
• Quality and quality spaces

Common application: knowledge-graph quality gates validate ontological consistency:

• TBox coherence (no unsatisfiable concepts)
• ABox consistency (instances satisfy TBox constraints)
• Schema-instance alignment

1.3 Formal Reasoning Systems

| System | Approach | Completeness | Use | |---|---|---|---| | Tableaux | Model construction attempt | Complete for DL | Ontology reasoning | | Resolution | Refutation | Complete for FOL | Theorem proving | | Sequent calculus | Structural proof | Complete for FOL | Proof theory | | Natural deduction | Introduction/elimination | Complete for FOL | Human-like proofs | | Type theory | Curry-Howard | ??? (depends on system) | Lean 4 foundation |

---

Part 2 — Neuro-Symbolic Integration

2.1 Integration Spectrum (Kautz 2020)

| Level | Description | Example | Pipeline role | |---|---|---|---| | 1 | Symbolic ← Neural | Neural generates symbolic output | Experience → symbols | | 2 | Symbolic → Neural | Symbolic knowledge guides neural | Structuring → embedding | | 3 | Hybrid pipeline | Neural + symbolic sequential | Full Project pipeline | | 4 | Tightly coupled | Neural and symbolic share representation | Graph of Thoughts | | 5 | Unified | Single system with both capabilities | Target architecture |

2.2 Key Architectures

• Neural theorem proving: Neural guide for tactic selection (GPT-f, AlphaProof)
• Knowledge graph embeddings: TransE, RotatE, CompGCN — vector representations of KG
• Graph neural networks: Message passing on knowledge structures
• Neurosymbolic concept learner: Learn visual concepts as logical programs
• Logic tensor networks: Differentiable first-order logic

2.3 Typical Neuro-Symbolic Position

Many verification-oriented pipelines operate at Level 3-4:

• Ingest phase: neural perception, extraction, or embedding.
• Articulation: neural-to-symbolic conversion from natural language to structured claims.
• Structuring: symbolic knowledge graph construction and validation.
• Consolidation: hybrid formal verification plus neural synthesis.
• Application: symbolic-to-neural feedback from validated knowledge to new tasks.

---

Part 3 — Commonsense Reasoning

3.1 Commonsense Knowledge Types

| Type | Description | Formalization Challenge | |---|---|---| | Physical | Objects fall, liquids flow | Qualitative physics, spatial reasoning | | Social | People have intentions, emotions | Theory of mind, folk psychology | | Temporal | Events have duration, causation | Allen's interval algebra | | Taxonomic | Dogs are animals | Inheritance hierarchies | | Default | Birds fly (unless penguin) | Nonmonotonic logic | | Causal | Pushing causes movement | Causal models (Pearl) |

3.2 Formal Approaches

• Circumscription (McCarthy): Minimize abnormality
• Default logic (Reiter): Default rules with exceptions
• Answer set programming: Stable models with negation-as-failure
• Probabilistic logic: Weight rules by confidence
• Large language models: Implicit commonsense from training data

3.3 Pipeline Integration

Commonsense reasoning commonly enters structured-AI pipelines in:

• Articulation: Interpreting tacit knowledge requires commonsense context
• Structuring: Default rules for knowledge graph completion
• Quality gates: Commonsense consistency checks on extracted knowledge
• Trust dynamics: Social commonsense for multi-agent coordination

---

Part 4 — Knowledge Graph Reasoning

4.1 Reasoning Tasks

| Task | Input | Output | Method | |---|---|---|---| | Link prediction | $(h, r, ?)$ | Missing tail entity | Embedding + scoring | | Type inference | Entity features | Entity type | Classification | | Rule mining | KG triples | Horn rules | AMIE, AnyBURL | | Query answering | Complex query | Answer entities | Query embedding | | KG completion | Partial KG | Complete KG | Combination methods | | Consistency checking | KG + constraints | Violations | DL reasoning |

4.2 Formal Quality Metrics

| Metric | Formula | Interpretation | |---|---|---| | Precision | TP / (TP + FP) | Fraction of predicted links that are correct | | Recall | TP / (TP + FN) | Fraction of true links predicted | | MRR | $\frac{1}{|Q|}\sum{q \in Q} \frac{1}{\text{rank}q}$ | Mean reciprocal rank of correct answers | | Hits@K | Fraction of correct in top-K | Retrieval quality | | Semantic validity | Ontology-consistency rate | Structural soundness |

4.3 Knowledge Graph Architecture

Experience → Raw triples (LLM extraction)
  → Articulation → Candidate assertions with confidence
    → Structuring → Validated KG with ontological constraints
      → Consolidation → Proven properties (Lean formalization)
        → Innovation → Novel inferences from verified base

Quality gates at each transition enforce:

• Confidence thresholds (configurable per domain)
• Ontological consistency (TBox reasoning)
• Provenance tracking (W3C PROV-O)
• Temporal validity (assertions expire)

---

Part 5 — Host-Repository Lean Extension Points

Do not assume repository-local Lean modules, tactics, or namespaces exist unless the host repository explicitly provides them. When a repository has local knowledge-graph or quality-gate modules, map them to this generic checklist:

| Local extension point | Symbolic AI aspect | Typical structures | |---|---|---| | Provenance / audit trail module | DAG well-formedness, trust composition | stages, edges, well-formedness predicates | | Causal / knowledge-graph module | Causal DAG and graph formalization | causal links, typed edges, confidence scores | | Quality-gate module | Predicate logic on quality measures | gates, thresholds, monotonicity theorems | | Classification / regime module | Decision procedures for regimes | regime types, decidability, monotonicity | | Causal-reasoning skill handoff | Causal DAGs and counterfactual reasoning | delegate to @lean-causal-reasoning |

---

Part 6 — Research Connections

6.1 Epistemic Mapping Targets

| KK (Known) | KU (Gap) | UU (To Discover) | |---|---|---| | DAG formalization | Full DL reasoning in Lean | OWL-to-Lean translation | | Propositional gates | Nonmonotonic formalization | Commonsense integration | | Trust composition | Neuro-symbolic verification | LLM-generated proof verification | | Provenance chains | Temporal knowledge evolution | Ontology evolution formalization |

6.2 Key References

• Brachman & Levesque (2004) — Knowledge Representation and Reasoning
• Hitzler et al. (2020) — Neuro-Symbolic AI: state of the art
• Marcus & Davis (2019) — Rebooting AI (commonsense perspective)
• Hogan et al. (2021) — Knowledge Graphs (comprehensive survey)
• Davis (2015) — Commonsense reasoning: an event calculus perspective

Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

• Author: r-irbe
• Source: r-irbe/proof-skills
• License: Apache-2.0
• Homepage: https://github.com/r-irbe/proof-skills

Install and usage instructions live in the source repository linked above.